Variation on a theme

Instead of adding the last two numbers together (as in the standard Fibonacci numbers), let's give ourselves a choice:
either add the current number to its parent or to its grandparent:

The Pseudo-Fibonacci numbers1 are formed by starting with

P1=1       P2=2 and       P3 =3

and thereafter, forming the next number in sequence with a choice:

either Pn= Pn-1+ Pn-2     --   called a right branch

or Pn'= Pn-1+ Pn-3          --   called a left branch

You can explore, yourself, a bit, by clicking on the buttons below:

        
P1:     P2:     P3: 
fast slow

For more about these, take a look here

1 Relatives of the Fibonacci numbers, these are also known as "Dailey numbers," after their discoverer.